Arithmetic/algebraicity criterion for power-series solutions of algebraic differential equations

Establish the equivalence, for the unique formal power series solution f(z)=∑_{n≥0} a_n z^n∈Q[[z]] to the non-linear algebraic differential equation f^{(n)}(z)=g(z,f(z),f'(z),…,f^{(n−1)}(z)) with g∈Q(z,y_0,…,y_{n−1}) defined at (0,t_0,…,t_{n−1})∈Q^{n}, between (i) algebraicity of f(z) over Q(z); (ii) integrality of its Taylor coefficients (existence of N with a_i∈Z[1/N] for all i); and (iii) ω(p)-integrality (existence of a function ω(p) with lim_{p→∞} ω(p)/p=∞ such that the initial ω(p) coefficients a_0,…,a_{ω(p)} lie in Z_{(p)} for every prime p).

Background

This conjecture extends classical arithmetic criteria for algebraicity of solutions from linear ODE to possibly non-linear algebraic differential equations. It strengthens and generalizes the spirit of the Grothendieck–Katz p-curvature conjecture by using integrality patterns in the Taylor expansion at a nonsingular point.

The authors note that Eisenstein’s theorem implies (i)⇒(ii) and that (ii)⇒(iii) is immediate, but the full equivalence is unknown in general and is the target of the conjecture.